Question: Solve for $x$ : $ 4|x - 9| - 8 = 2|x - 9| + 7 $
Subtract $ {2|x - 9|} $ from both sides: $ \begin{eqnarray} 4|x - 9| - 8 &=& 2|x - 9| + 7 \\ \\ { - 2|x - 9|} && { - 2|x - 9|} \\ \\ 2|x - 9| - 8 &=& 7 \end{eqnarray} $ Add ${8}$ to both sides: $ \begin{eqnarray} 2|x - 9| - 8 &=& 7 \\ \\ { + 8} &=& { + 8} \\ \\ 2|x - 9| &=& 15 \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{2|x - 9|} {{2}} = \dfrac{15} {{2}} $ Simplify: $ |x - 9| = \dfrac{15}{2}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 9 = -\dfrac{15}{2} $ or $ x - 9 = \dfrac{15}{2} $ Solve for the solution where $x - 9$ is negative: $ x - 9 = -\dfrac{15}{2} $ Add ${9}$ to both sides: $ \begin{eqnarray} x - 9 &=& -\dfrac{15}{2} \\ \\ {+ 9} && {+ 9} \\ \\ x &=& -\dfrac{15}{2} + 9 \end{eqnarray} $ Change the ${ + 9}$ to an equivalent fraction with a denominator of $2$ $ x = - \dfrac{15}{2} {+ \dfrac{18}{2}} $ $ x = \dfrac{3}{2} $ Then calculate the solution where $x - 9$ is positive: $ x - 9 = \dfrac{15}{2} $ Add ${9}$ to both sides: $ \begin{eqnarray} x - 9 &=& \dfrac{15}{2} \\ \\ {+ 9} && {+ 9} \\ \\ x &=& \dfrac{15}{2} + 9 \end{eqnarray} $ Change the ${ + 9}$ to an equivalent fraction with a denominator of $2$ $ x = \dfrac{15}{2} {+ \dfrac{18}{2}} $ $ x = \dfrac{33}{2} $ Thus, the correct answer is $x = \dfrac{3}{2} $ or $x = \dfrac{33}{2} $.